It makes me deeply uncomfortable to use a public domain and get public certificates for something inherently so private
You can obviously run your own CA, great exercise but why? What really makes you that uncomfortable? Once you go with the wildcard nobody will know about your internal hosts or whatever. Even if the domain is taken down, you’re offline or wtv your local DNS server will still be able to serve replies to those internal subdomains. You don’t need to publish those subdomains (A records) in a public DNS server, just on your own internal DNS server.
I guess if you rally want to take the CA route those tools I provided before are the best option. Simply issuing a certificate (without a CA) and allowing it on a browser might also work for you - less risks of stolen PK as described.